Exponential decrease in grain size with linear distance of sediment transport is expressed as a variation of Sternberg's Law. This variation is Y = Yoe-ax in which Y0 is the initial diameter of a particle, Y is the diameter of the particle after Travelling a distance X, and a is the slope of the curve. This slope was designated the coefficient of size reduction by Sternberg, Determination of paleoslope attitude, and paleocurrent direction, and sedimentary anisotropy were achieved from the field measurements on vectoral properties of foreset beds and current ripples, and from the examination of sedimentologic-stratigraphic maps such as grain size distribution, isopachs, and facies.
The basic equation (Y = Yoe-ax) is applied to grain diameters of sediment samples from Arctic rivers, thus representing sedimentation on the modern temporal plane. Next, the mathematical operations carried out on both scalar and vectoral entities are applied to the upper and lower parts of a Silurian member (the Grimeby Sandstone in the Niagara Peninsula) of Ontario, in order to illustrate the persistence of the exponential law through a small interval of geologic time. Superposition of the size-distance curves representing top and bottom beds show parallelism of slope. The operations applied across two members (the Grimsby and overlying Thorold sandstone) show a similar parallelism of size-distance curves. The operations applied across several formations representing almost an entire geologic period (the Triassic sandstones of northeastern British Columbia - Toad, Liard, and Grey Beds) yield a family of negative, exponential, size-distance curves, drawn from the textural analyses. Finally, the operations are applied to formations representing a long interval of geologic time (the conglomerates of the lower Mississippian Pocono and lower Pennsylvanian Pottsville). Size-distance curves based on textural variations along a sampling line trending northwesterly across Pennsylvania were constructed and superposed on the same graph. The resulting relationship demonstrates that under prograding conditions a natural law of growth for sedimentary clastic bodies exists and persists over long periods, being expressed in the form of a family of negative exponential curves. Also, this law together with sedimentary anisotropy and progradation constitute a sedimentologic continuum operating through this different but successive interval of geologic time.